Problem 51625. Solve an ODE: diffusion problem 2

ChrisR

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Problem statement
In the solution of a problem involving diffusion, the following ordinary differential equation arises
 
where a is a positive constant and primes denote differentiation with respect to the independent variable η. The function f and its derivatives vanish at infinity, and it is subject to the constraint
Write a function to solve this problem—that is, return values of f at specified values of η. 
Background
The physical problem involves diffusion of a quantity instantaneously injected at a point—for example, a spill of a contaminant in an initially clean river. The ODE results from transforming the diffusion equation—a partial differential equation in time and a spatial coordinate—with a similarity solution. 